A Inquiry regarding a matrix tensor notation...

  • Thread starter berlinspeed
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Summary
When comparing ##g_{\mu\nu}## with ##\eta_{\mu\nu}## in curved spacetime.
So if ##P_{0}## is an event, and I have ##\mathcal {g_{\mu\nu}(P_{0})}=0## and ##\mathcal {g_{\mu\nu,\alpha\beta}(P_{0})}\neq0##, does this notation mean ##\partial\alpha\partial\beta## or simply ##\partial(\alpha\beta)##? And what is the significance of it? Why can't it be zero in curved spacetime?
 

Cryo

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How about defining your terms? All of them? Also if you want to talk about curved space-time, please specify which derivatives you are talking about. Simple partial derivatives, covariant derivatives, if covariant, are you using Levi-Civita connection. etc
 
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if ##P_{0}## is an event, and I have ##\mathcal {g_{\mu\nu}(P_{0})}=0##
This makes no sense; the metric never vanishes. I think what you meant to write here is ##\mathcal {g_{\mu\nu}(P_{0})}=\eta_{\mu \nu}##.

does this notation mean ##\partial\alpha\partial\beta## or simply ##\partial(\alpha\beta)##?
The notation ##\mathcal {g_{\mu\nu, \alpha \beta}(P_{0})}## means ##\mathcal {\partial_\alpha \partial_\beta g_{\mu\nu}(P_{0})}##. I have no idea what you mean by ##\partial\alpha\partial\beta## or ##\partial(\alpha\beta)##.
 
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what is the significance of it?
What is the significance of what?

It would help if you would give a specific reference for where you are getting this from. Also it would help to have some idea of how much background you have in GR; you marked this thread as "A" level but the questions you are asking don't indicate that you have that level of background knowledge.
 

Orodruin

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You can always find a coordinate system such that ##g_{\mu\nu} = \eta_{\mu\nu}## and the Christoffel symbols vanish at a given point ##p##. However, in such a coordinate system, the Riemann curvature tensor at ##p## takes the form
$$
R^d_{cab} = \partial_a \Gamma^d_{bc} - \partial_b \Gamma^d_{ac}.
$$
Inserting the explicit form of the Christoffel symbols and lowering the ##d##-index leads to
$$
R_{abcd} = \frac{1}{2}(g_{bc,ad} + g_{ad,bc} - g_{bd,ac} - g_{ac,bd}).
$$
Therefore, if ##g_{\mu\nu,\alpha\beta} = 0##, then the curvature tensor would be identically equal to zero at ##p##, i.e., the manifold would be flat at ##p##, contrary to the assertion that the manifold is curved at ##p##.
 
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You can always find a coordinate system such that ##g_{\mu\nu}## and the Christoffel symbols vanish at a given point ##p##.
Such that ##g_{\mu \nu} = \eta_{\mu \nu}## and the Christoffel symbols vanish at a given point ##p##. The metric ##g_{\mu \nu}## does not vanish.
 

Orodruin

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Such that ##g_{\mu \nu} = \eta_{\mu \nu}## and the Christoffel symbols vanish at a given point ##p##. The metric ##g_{\mu \nu}## does not vanish.
Meh, that is what happens when you edit your post too much before posting, you remove things that were needed to make it correct ...

Edit: Fixed ...
 
You can always find a coordinate system such that ##g_{\mu\nu} = \eta_{\mu\nu}## and the Christoffel symbols vanish at a given point ##p##. However, in such a coordinate system, the Riemann curvature tensor at ##p## takes the form
$$
R^d_{cab} = \partial_a \Gamma^d_{bc} - \partial_b \Gamma^d_{ac}.
$$
Inserting the explicit form of the Christoffel symbols and lowering the ##d##-index leads to
$$
R_{abcd} = \frac{1}{2}(g_{bc,ad} + g_{ad,bc} - g_{bd,ac} - g_{ac,bd}).
$$
Therefore, if ##g_{\mu\nu,\alpha\beta} = 0##, then the curvature tensor would be identically equal to zero at ##p##, i.e., the manifold would be flat at ##p##, contrary to the assertion that the manifold is curved at ##p##.
Thanks for the explanation however my question was simply regarding the notation of ##\mathcal {g_{\mu\nu,\alpha\beta}(P_{0})}\neq0## -- does the ##,\alpha\beta## signify ##\partial_{\alpha}\partial_{\beta}## or ##\partial_{(\alpha\beta)}##, now it seems to be the former..
 
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Orodruin

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Thanks for the explanation however my question was simply regarding the notation of ##\mathcal {g_{\mu\nu,\alpha\beta}(P_{0})}\neq0## -- does the ##,\alpha\beta## signify ##\partial_{\alpha}\partial_{\beta}## or ##\partial_{(\alpha\beta)}##, now it seems to be the former..
Your OP had three questions;
Summary: When comparing ##g_{\mu\nu}## with ##\eta_{\mu\nu}## in curved spacetime.

Why can't it be zero in curved spacetime?
I was answering this one as the others had been addressed.
 

Ibix

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Thanks for the explanation however my question was simply regarding the notation of ##\mathcal {g_{\mu\nu,\alpha\beta}(P_{0})}\neq0## -- does the ##,\alpha\beta## signify ##\partial_{\alpha}\partial_{\beta}## or ##\partial_{(\alpha\beta)}##, now it seems to be the former..
As noted above, ##g_{\mu\nu,\alpha\beta}## is a shorthand notation for ##\partial_\alpha\partial_\beta g_{\mu\nu}##, which in turn is shorthand for $$\frac{\partial}{\partial x^\alpha}\frac{\partial}{\partial x^\beta}g_{\mu\nu}$$You seem to me to be trying to ask if it might instead mean$$\frac{\partial^2}{\partial x^\alpha\partial x^\beta}g_{\mu\nu}$$But that's exactly the same thing.
 
As noted above, ##g_{\mu\nu,\alpha\beta}## is a shorthand notation for ##\partial_\alpha\partial_\beta g_{\mu\nu}##, which in turn is shorthand for $$\frac{\partial}{\partial x^\alpha}\frac{\partial}{\partial x^\beta}g_{\mu\nu}$$You seem to me to be trying to ask if it might instead mean$$\frac{\partial^2}{\partial x^\alpha\partial x^\beta}g_{\mu\nu}$$But that's exactly the same thing.
No I thought it was the product of ##\alpha## and ##\beta## but it's not, which wouldn't make much sense.
 

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